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agronomy
Review
Spatial Modelling of Within-Field Weed Populations;
a Review
Gayle J. Somerville 1, * , Mette Sønderskov 1 , Solvejg Kopp Mathiassen 1 and Helen Metcalfe 2
1 Department of Agroecology, Aarhus University, DK-4200 Slagelse, Denmark;
mette.sonderskov@agro.au.dk (M.S.); sma@agro.au.dk (S.K.M.)
2 Sustainable Agriculture Sciences, Rothamsted Research, Harpenden, Hertfordshire AL5 2JQ, UK;
helen.metcalfe@rothamsted.ac.uk
* Correspondence: gaylene.somerville@lincoln.ac.nz or gayesomerville@hotmail.com
Received: 23 June 2020; Accepted: 16 July 2020; Published: 20 July 2020
Abstract: Concerns around herbicide resistance, human risk, and the environmental impacts of
current weed control strategies have led to an increasing demand for alternative weed management
methods. Many new weed management strategies are under development; however, the poor
availability of accurate weed maps, and a lack of confidence in the outcomes of alternative weed
management strategies, has hindered their adoption. Developments in field sampling and processing,
combined with spatial modelling, can support the implementation and assessment of new and more
integrated weed management strategies. Our review focuses on the biological and mathematical
aspects of assembling within-field weed models. We describe both static and spatio-temporal
models of within-field weed distributions (including both cellular automata (CA) and non-CA
models), discussing issues surrounding the spatial processes of weed dispersal and competition
and the environmental and anthropogenic processes that affect weed spatial and spatio-temporal
distributions. We also examine issues surrounding model uncertainty. By reviewing the current
state-of-the-art in both static and temporally dynamic weed spatial modelling we highlight some
of the strengths and weaknesses of current techniques, together with current and emerging areas
of interest for the application of spatial models, including targeted weed treatments, economic
analysis, herbicide resistance and integrated weed management, the dispersal of biocontrol agents,
and invasive weed species.
Keywords: spatio-temporal models; integrated weed management; weed mapping; targeted weed
treatment; site specific weed management
1. Introduction
Weeds are the main source of yield loss in crops, causing up to 34% loss across agricultural
and horticultural crop production [1,2]. Farmers’ primary management target is maximising
economic returns, which for many farmers involves the simultaneous handling of multiple objectives
regarding weeds: reducing weed seedbank size, the eradication of competitive weeds, halting new
invasions, and combatting herbicide resistance (HR). Many weed species generate populations with a
heterogeneous density and distribution, which makes weed management difficult [3,4].
Site-specific management is already commonplace in many aspects of farming. Information-based
management systems to adapt fertiliser distribution across the field were first introduced in the
mid-1980s [5], and since then, precision farming techniques, including GPS steering, soil mapping,
and variable rate seeding are becoming increasingly popular. However, the uptake of site-specific
weed management is lacking, largely due to the unavailability of accurate within-field weed species
distribution maps. However, in recent years, developments in precision agriculture have led to an
Agronomy 2020, 10, 1044; doi:10.3390/agronomy10071044 www.mdpi.com/journal/agronomyAgronomy 2020, 10, 1044 2 of 18
improved understanding of within-field spatial distributions of weeds, which is a key to improving
the adoption rate of such management practices for weeds. Spatio-temporal models can predict the
local development of weed populations, which may be useful in mapping pre-emergent herbicide use,
although within-field distribution can be unpredictably variable [6]. The accurate modelling of static
within-field weed distributions can also be of benefit to farmers investigating herbicide resistance,
checking for invasions of new weed species, and implementing biological control agents. The spatial
heterogeneity within weed populations can be incorporated into temporal weed models, creating more
realistic simulations, and improving differentiated weed management [7]. Modelling within-field
distributions in this way not only benefits farmers by supporting site-specific weed management,
but such spatial models can also provide researchers with an improved understanding of the underlying
processes determining weed spatial distributions, and open up new avenues for research and
investigation to better understanding of the ecology and biology of agricultural weeds.
In this review, we outline the current state-of-the-art in both static and temporally dynamic weed
spatial modelling. We highlight some of the strengths and weaknesses of current techniques and
discuss the usefulness of current models. We also identify areas of emerging interest, where models
of within-field weed spatial distributions could be instrumental in the development of novel weed
management techniques.
2. Static Models of Within-Field Weed Distributions
Static within-field models of weed populations predict weed distribution and densities across fields.
They are either based on real-time recording of the location of individual weeds or sampling at a number
of discrete locations and interpolating the abundances at unsampled locations. Additional information
about biotic and abiotic factors affecting within-field weed distributions can be included to improve
the model predictions at unsampled locations.
2.1. Real-Time Weed Monitoring
Real-time approaches to weed management involve the assessment and treatment of weeds in the
same operation, or mapping for imminent treatment. This is in combination with a decision algorithm,
which allows an immediate decision to either treat, or not treat, individual parts of a field. Different
real-time approaches are already feasible, but are not yet at the stage of widespread commercialization,
and are continuously under development [8]. One common type of real-time approach involves optical
sensors mounted on sprayers, with real-time image analysis and decision making as the machinery
passes over the field e.g., [9–11]. Alternatively, cameras can be mounted on unmanned aerial vehicles
for rapid mapping of whole fields, and machine learning techniques can be employed to rapidly
produce spray maps for site-specific weed management [12,13]. The image processing methods
deployed in these real-time approaches generally involve segmentation of the image to locate areas
that are different from the crop. The sophistication of these algorithms is constantly being developed
to achieve better classification of vegetation, as current methods are largely limited to identifying any
vegetation growing prior to crop establishment, or outside the emerged crop rows [14]. The detail of
the maps produced by these real-time monitoring methods is unrivalled by other lower-tech mapping
options, however, both the cost associated with the equipment required to produce such maps, and the
dynamic nature of the research, are barriers to uptake for many farmers.
2.2. Manual Sampling of Within-Field Weed Distributions
In addition to real-time assessment, field-wide weed maps can also be created from manually
collected data of weed distributions. However, manual collection is less often used in practice, as it is
more time-consuming than real-time monitoring techniques. Rew and Cousens [15] outlined three
groups of methods for manual sampling of weeds: discrete area sampling, continuous area sampling,
and co-ordinate mapping of individual plants. The three methods have pros and cons: discrete area
sampling consists of point observations (usually weed counts in pre-selected quadrats) and is relativelyAgronomy 2020, 10, 1044 3 of 18
cheap and fast to conduct, yet leaves a large part of the field unsampled; continuous area sampling
takes observations across the whole fields assessing average densities across pre-defined grid cells,
and requiring observer consistency in their assessment of weed densities; and co-ordinate mapping
provides highly detailed maps, but requires very intensive sampling efforts.
2.3. Creating Maps
After sampling, interpolation is required to estimate weed densities in the spaces between
sampling points. Several interpolation methods exist [16], for creating smoothed maps of each weed
species’ distribution, with Kriging proving a particularly popular method for producing unbiased
estimates [17]. The success of any interpolation method will depend on the quality of the data obtained.
Maps based on observations at coarse resolutions will be poor in detail, yet sampling at fine resolution
may highlight unnecessary details, and prevent the observation of any broader patterns [18]. In contrast
to a wide variety of smoothed maps, density maps maintain separated zones within previously defined
boundaries [16]. In density maps, individual samples are either kept separate (with one sample
per zone) or combined locally. This creates maps with changes in density occurring abruptly, at the
boundaries between zones [19,20].
Within-field weed distributions can be relatively stable between years [21] making any map produced
useful in future years, although annual weeds are more variable in distribution [22]. Patch stability does,
however, not necessarily imply stability in weed density [23]. The greater the time between sampling and
prediction, the less accurate the prediction becomes. Where the factors affecting patch stability are known,
it is possible to apply buffer zones around mapped weed patches to account for potential spread [15].
2.4. Improving Static Weed Distribution Models
In some cases, it is possible to improve the reliability of a static weed distribution model by
incorporating supplementary data. The inclusion of environmental properties that have known
associations with a particular weed species distribution can be a valuable addition to the development
of static weed distribution models, particularly where sampling of weeds is sparse. Co-kriging is one
method of interpolation that allows the use of maps of environmental properties, for example, soil
type, to be used to supplement the weed data and create a more robust map of weed distribution [24].
However, the strength of association between weeds and such environmental properties can be
scale-dependent [25], and should be considered in model development. The best mapping strategy will
vary between different crops and weed management systems, influenced by the quality and quantity
of the available data, as well as the purpose of the map.
3. Spatio-Temporal Models of Within-Field Weed Distributions
Spatio-temporal models not only aim to map the within-field distribution of weeds, but add
predictions of weed development over time and space. Their parametrisation requires knowledge
about initial weed distribution, as well as a detailed understanding of ecological, environmental,
and management practices, in order to simulate the complete life cycle of the plant. These models are
therefore more costly in terms of parameterisation than static weed models, but have the potential to
be more valuable in terms of their wider use across a range of applications.
When building spatio-temporal models at the field scale, the size of the field and the number
of weeds becomes important. Each square metre of a field can contain thousands of weed seeds
resulting in millions of weed seeds in even the smallest fields [26]. From each emerging weed plant,
there are spheres of influence spreading out in terms of competition and dispersal, which, at the
field scale, creates billions of interactions. The number of interactions is one of the main reasons that
spatio-temporal modelling is infrequently used. Spatio-temporal models deal with these difficulties in
different ways.Agronomy 2020, 10, 1044 4 of 18
Agronomy 2020, 10, x FOR PEER REVIEW 4 of 18
3.1. Cellular Automata (CA) Models
3.1. Cellular spatio-temporal
Typically, Automata (CA) Models
models use a cellular automata (CA) approach. CA models utilise a
lattice-based map of
Typically, plant locationsmodels
spatio-temporal (examples
use ashown
cellularin Figure 1).
automata These
(CA) modelsCA
approach. either simulate
models utilise single
a
lattice-based map of plant locations (examples shown in Figure 1). These models either
dynamic entities confined to one grid point of the lattice, sub-classified here as CA-E models (Figure 1A), simulate
singlesubpopulations
or utilise dynamic entitieswithin
confined
eachto cell
one ofgrid
thepoint of the
lattice, lattice, sub-classified
sub-classified here as
here as CA-P CA-E models
models (Figure 1B).
(Figure 1A), or utilise subpopulations within each cell of the lattice,
In a CA-E model, a maximum set number of entities is confined to the number of points sub-classified here aswithin
CA-P the
models (Figure 1B). In a CA-E model, a maximum set number of entities is confined to the number
lattice (81 in the case of Figure 1) with competition and dispersal of pollen, seed, and other vegetative
of points within the lattice (81 in the case of Figure 1) with competition and dispersal of pollen, seed,
material assessed between nearby points in the lattice. In CA-E models information concerning
and other vegetative material assessed between nearby points in the lattice. In CA-E models
vegetation and environment at each point can be quite complex, but any plants growing between the
information concerning vegetation and environment at each point can be quite complex, but any
points are not
plants specifically
growing betweenassessed.
the pointsIn CA-P
are notmodels, inter- assessed.
specifically and intraspecific
In CA-Pcompetition
models, inter-is confined
and
within each subpopulation within a lattice cell, with the dispersal of reproductive
intraspecific competition is confined within each subpopulation within a lattice cell, with material occurring
the
primarily within
dispersal and between
of reproductive nearby
material sub-populations.
occurring primarily withinIn CA-Pand models
between all plants
nearby within the lattice,
sub-populations.
of theInspecies underall
CA-P models consideration, are lattice,
plants within the assessed.
of the species under consideration, are assessed.
(A) (B)
Figure
Figure 1. Visual
1. Visual representation
representation of of examples
examples ofofcellular
cellularautomata
automata (CA)
(CA) models
modelsusing
usingeither
either(A)(A)points
points in
in amodel
a CA-E CA-E model or (B)incells
or (B) cells in a CA-P
a CA-P model.model.
In eachIn each
case case a single
a single population
population of weeds
of weeds is analysed
is analysed across
across a 9 × 9 lattice. In the CA-E model (A), there is typically one plant measured at
a 9 × 9 lattice. In the CA-E model (A), there is typically one plant measured at each point, whereas in each point,
whereas
the CA-P in the
model CA-P
(B), theremodel (B), there defined
are spatially are spatially defined sub-populations
sub-populations containingcontaining
any number any from
number
zero to
from zero to their maximum capacity of weeds, and seeds. In both examples, weeds at the central
their maximum capacity of weeds, and seeds. In both examples, weeds at the central yellow position
yellow position can disperse their seeds or pollen to the points or cells indicated by the circle.
can disperse their seeds or pollen to the points or cells indicated by the circle.
3.1.1.3.1.1.
CA CA Lattice
Lattice
Lattice cells may be any shape which tessellates, although squares and hexagons are most
Lattice cells may be any shape which tessellates, although squares and hexagons are most common.
common. The cell shape influences the type of dispersal algorithms used within the lattice. Squares
The cell shape influences the type of dispersal algorithms used within the lattice. Squares simplify the
simplify the incorporation and examination of anthropogenic dispersal, which is often along crop
incorporation
rows [27–29], and examination
whereas hexagons ofsimplify
anthropogenic dispersal, which
the parameterisation is often
of natural along
dispersal, cropisrows
which [27–29],
typically
whereas
radial [30]. Brix and Chadoeuf [31] combined aspects of both CA-E and CA-P design with smaller [30].
hexagons simplify the parameterisation of natural dispersal, which is typically radial
Brix cells
and placed
Chadoeuf within[31] combinedof
a framework aspects of bothtoCA-E
larger shapes, anddistribution
simulate CA-P design andwith smallerover
germination cellsone
placed
within a framework
growing season. of larger shapes, to simulate distribution and germination over one growing season.
WithinWithin
CA-P CA-P models,
models, cellsize
cell sizedetermines
determines the size
size ofofeach
eachsub-population,
sub-population, imposing
imposingartificial
artificial
constraints on the population. Competition is confined within cells and affects all plants within each
constraints on the population. Competition is confined within cells and affects all plants within each cell,
cell, both
whereas whereas both anthropogenic
anthropogenic and natural anddispersal
natural occur
dispersal occur cells,
between between cells, relocating
relocating some of
some percentage
percentage of seeds/pollen to each newly invaded cell. The dispersal of seeds typically
seeds/pollen to each newly invaded cell. The dispersal of seeds typically declines rapidly with distance, declines
rapidly with distance, meaning that species with limited dispersal are best replicated using smaller
meaning that species with limited dispersal are best replicated using smaller cell sizes, in order to achieve
cell sizes, in order to achieve annual changes in distribution [32]. However, larger cell sizes also have
annual changes in distribution [32]. However, larger cell sizes also have advantages: since competition is
typically confined to within individual cells, they need to contain enough crop plants to capture the effectAgronomy 2020, 10, 1044 5 of 18
of each weed on crop growth; larger cells also allow a larger area to be simulated with the same number
of sub-populations, which decreases the runtime of the model. Cell size choice may also need to consider
anthropogenic processes, e.g., if combine harvester trails are simulated, cell width must be related to
trail width [27,28]. Depending on the simulated situation, larger cells can be used without affecting the
results, although some details on spatial differences within the weed population will be eliminated [33].
The realistic spatial containment of competition and biological dispersal imposed by cell size are two of
the main advantages of spatial modelling of within-field weed populations.
3.1.2. Edge of Lattice
Spatial models are highly variable in how they deal with the edge of the lattice, and how the
chosen dispersal functions will receive and deposit biological material outside the lattice boundary.
However, edge effects are less important when modelling spread from a central point [33].
The simplest assumption is that the edges of the lattice neighbour an external area, which can be
biologically different to the population within the lattice. This external area is typically uniform, and has
been modelled as non-responsive [34], or barren [30]. However, in these studies, sub-populations near
the edge of the lattice suffer a comparative genetic, or seed deficit, when compared to more centrally
located sub-populations, which are both donating, and receiving, pollen and seed. Edge effects can be
minimised by using more detailed lattices, [30] or by using lattices covering a larger area [27].
Edge effects can be accounted for by reflecting discarded seed and pollen from cells near the edges
back into the cell it originated from, like placing a mirror along the edges of the lattice. Another method
is to metaphysically wrap the edges of the lattice. When wrapping is used, weeds at one edge of the
lattice (position ‘V’ in Figure 1) are considered to neighbour weeds on the opposite side of the lattice
(position ‘W’ in Figure 1). Wrapping means that loss (and gain) from cells near the edge of the lattice
is eliminated [33,35]. However, fields with directional movement, or spatial zones with gradients,
may be unsuitable for wrapping.
3.1.3. Spatial Replication and Subpopulations
Early CA-E modelling using a lattice layout was conducted by Rees and Paynter [35], who simulated
weed growth and reproduction over a square lattice (of 5625 sites). Their simulation allowed for
only a single weed plant (absent or present with a variety of growth stages) at each site position.
More complex CA-E models have been used recently, with defined levels and types of weed flora
possible at each CA-E point [36,37]. Richter, et al. [38] presented an early CA-P model by using
matrix population modelling within each sub-population. In contrast, most of the more recent work
using CA-P modelling utilise sub-populations in the CA-P format, [27,32], with the added complexity
of 3-D soil profiles in Metcalfe, et al. [39]. The use of sub-populations in CA-P simulations means
that every weed within the lattice is accounted for, each with its own genetic, germination, survival,
and seed set probabilities. The multitude of calculations within CA-P models can be simplified by
using matrix population models [38], or by applying various probability theories within and between
sub-population groups. In CA-E modelling, calculations do not involve individual plants, but instead
examine changes in entity state. The use of states in CA-E models means less detail is included about
genetics and individual plants, but that different species can be more easily included. In CA-E models,
field surveys can be directly matched to the grid points, removing the need to interpolate weed maps
for initial model parameterisation.
Seeds typically disperse over smaller distances than pollen, which can be simulated well in spatial
models. However, the concept of cells limits the realism, by defining biological dispersal to a series of
artificial ‘jumps’ between cells. This issue is more important for species with smaller seed distributions,
and necessitates the use of smaller cells, however, results can still realistically mimic field data [4].
These rigid boundaries between cells can be avoided with the use of either partial differential equations,
or directional geometry in non-CA modelling [40,41]. In any case, the containment of seed dispersal
possible in a spatial model means that weed population growth is realistically slower, due to localisedAgronomy 2020, 10, 1044 6 of 18
intra-species competition. In comparison, in non-spatial weed models intra-species weed competition
does not affect early population growth [33,34].
3.2. Non-CA Models
CA models are particularly useful for simulations in 2-D over a reasonably homogeneous environment
such as a crop field. However, in some cases, the rigid nature of the CA lattice and the uniform nature
of the utilised environment may be either unnecessary, or insufficient to provide optimal simulations.
For example, Andujar et al. [41] utilised both directional triangular geometry and directional rectangular
geometry, to model a new invasive weed dispersing in individual GPS guided harvester trails.
Diverse landscapes can be divided into subdomains in non-CA spatial models, with dispersal
between individual subdomains calculated in a similar way to dispersal between the points in a CA
framework. However, the use of distinct sub-domains is more useful at the landscape level. By avoiding
a CA lattice and instead working with sub-domains, it is possible to include separate calculations for
each inter-subdomain pollen and seed exchange, with the inclusion of individual terms to account
for agronomic and environmental conditions, which can be unique to each inter-subdomain exchange.
Whereas this system could be used for within-field gradients of environmental variables, it is more
commonly used in inter-field dispersal studies where larger and more predictable variation occurs [40,42].
Other non-CA models work at the level of individual plants, monitoring the location of each plant
within the field and determining the zone of influence each plant has on its neighbours. Colbach et al. [43]
implemented such a model using 3D cylindrical shapes to represent individual plants.
4. Modelling Spatial Processes
Dispersal and competition processes affect weed distribution within a field, with the level of
their influence varying in both space and time. Spatio-temporal modelling methods have been
designed for dispersal and competition, both as independent spatial processes, and as components of
spatio-temporal population models.
4.1. Dispersal
Seed dispersal is an important aspect of weed dispersal and of primary importance in spatial
modelling, as seeds have a high capacity to colonise new areas [44]. In addition, seed dispersal is
one of the main causes of spatial heterogeneity in fields [45]. Seed dispersal range determines the
possibility of new site colonisation and depends on the weed species and seed characteristics, as well
as external factors, such as wind speed and direction. Modelling the spread of weeds within fields is
achieved by predicting the proportion of plant material (typically seeds), produced at one location,
that moves to another location in a given time period, at the field scale.
Increasing wind speed can increase the number of seeds dispersed and the distance of dispersal [46].
The most common distribution is a leptokurtic pattern with a maximum close to the seed producing
plants. This implies that for many weed species natural seed dispersal occurs over short distances,
making patch formation likely. Early maturing weed species will shed seeds prior to crop harvest [47].
The distance of dispersal depends partly on the germination pattern and maximum height of the parent
weed plants as well as physical properties of the seeds. Dispersal of seeds and pollen in the direction of
crop rows can be due to natural dispersal in tall crops, where they move more freely in the row direction
compared to movement across the crop rows [23]. In shorter crops, where the weed flowers are above
the crop canopy, natural dispersal is likely to extend further, and be more uniform [29,48]. In addition,
vegetative material can be inadvertently dispersed by the farmer, whist conducting farm management
activities. Soil tillage can displace seeds and vegetative material horizontally in space [49] and
redistribute them among soil layers, according to depth and type of cultivation [50]. Seeds remaining
on the weed plants at crop harvest are dispersed by the combine harvester. The proportion of seeds
entering the harvester depends on the harvest height, phonological development of the species,
harvesting date, and characteristics of the propagules [51]. Dispersal parameters had more influenceAgronomy 2020, 10, 1044 7 of 18
on the rate of spread than demographic parameters, and increased up to 16-fold when seeds were
dispersed by the harvester [23,52–54].
In contrast to seed dispersal, pollen has been tracked over greater distances, although most plants
are pollinated by close neighbours [55]. Pollen movement disperses genetic material within and between
populations, thereby decreasing genetic difference in the overall population. Pollen dispersal does not
structurally relocate weeds, but only disperses genetic material within a population. The incorporation of
pollen dispersal is therefore only necessary in spatio-temporal models, which study genetic change in
terms of movement and mixing of plant genes. It has been used to simulate dispersal and mixing of GM
genes, and herbicide resistant genes [28,56]. In contrast, in a non-spatial temporal model, all genes can mix
freely from the first instance, making it impossible to accurately compare genetic changes within a field.
4.2. Competition
Competition is a major factor in weed population dynamics within a field, where plant densities
affect both weed reproductive rates and crop yield. Competition modelling typically includes two-way
interactions, influenced by both interspecific competition between the crop and weed and among
weed species, and intra-specific competition within crop and weed species. The patchy nature of weed
distributions will affect within-field competition modelling.
The modelling of intraspecific competition in monocultures has been described using a reciprocal
function [57,58], but intraspecific competition is largely neglected in competition modelling [59].
Interspecific competition has been modelled based on two underlying concepts: replacement series
and additive design. In replacement series, the density is constant, and the composition of species are
varied. In the additive design, both the density and the proportion of species changes. The value and
application of these designs both have merits, depending on the situation to be modelled e.g., [60,61].
The hyperbolic yield loss function proposed by Cousens [62] belongs to additive design, and has been
extensively used and adjusted to a range of purposes. While the basic function models the effect of only
one weed species on the crop yield, later versions include crop density, emergence time, and several
weed species [63–65]. The above models can be considered as descriptive or empirical models, contrary
to mechanistic or explanatory modelling approaches. The latter often uses competition for light as the
limiting resource, considering canopy structure and the space plants occupy. Four such models were
compared by Deen et al. [66].
The modelling of competition within spatio-temporal models typically utilises previously
published competition parameters and equations to quantify competition. In spatio-temporal models,
localised competition can capture weed patchiness, thereby improving accuracy compared with weaker
field-wide competition in non-spatial temporal models. The extrapolation of localised competition
equations to field-wide competition becomes less reliable when the calculated relationship between
density and competition is not linear.
5. Environmental and Anthropogenic Processes
Many biotic, abiotic and anthropogenic processes have been incorporated into, and investigated
with, various non-spatial weed models [59,67]. These processes can also be investigated with spatial
models with the added advantage that all input parameters can be incorporated in ways which are
non-uniform [30], randomly dynamic, or even locally responsive [37]. However, the use of spatial models
for weed populations without spatial differentiation in either the weed distribution or genetics, or in the
environmental or agronomic practices, may be of little advantage over using non-spatial models [30].
Environmental and anthropogenic dispersal of reproductive material was discussed in Section 4.1.
5.1. Environmental
Environmental factors including light, precipitation, temperature, soil type, pH, nutrients,
and moisture can vary within fields, which may influence the location of weed patches [68]. Seeds landing
in unsuitable environments are unlikely to germinate, or to compete successfully. Spatial heterogeneity ofAgronomy 2020, 10, 1044 8 of 18
weed populations can therefore be partially determined by the local distribution of sites with suitable
abiotic conditions [45]. Soil texture will influence seed burial [69], and germination; for example
Galium aparine and Alopecurus myosuroides are associated with a larger clay content, whereas Senecio vulgaris
thrives on lighter soil [70,71]. Other soil variables such as carbon, water, and macronutrient levels
have also been linked to weed distributions [70], although these are not well understood for many
species [72]. Metcalfe et al. [39] developed a dynamic model of the within-field spatial distribution
of A. myosuroides, which incorporated within-field spatial variation in several environmental variables,
including topography and soil. They demonstrated that incremental changes to the life cycle of the species
due to these environmental properties, when combined in a spatio-temporal model, resulted in realistic
simulations of within field weed distributions.
5.2. Anthropogenic
Weed distribution and dispersal are also influenced by annual activities in the cropping system.
Any spatial variability in current agronomic practices within the field will influence the germination,
growth, and survival of weeds [73]. For example, fertiliser application, crop orientation, and crop
establishment rate may vary across the field, and can affect weed composition and fitness [70,74,75].
Herbicide application and efficiency may also vary within a field, due to drift, soil properties, or crop
residues, which can influence weed spatial distribution within a field [76–78]. In addition, weed
detection and cross pollination can be affected by temporal differences in plant growth patterns.
When suitable bare weed habitat is transient this can affect patch colonisation, particularly when seed
dispersal is pulsed, rather than continuous [79].
5.3. Field Edges
The field-edge is known to represent a distinct habitat within the field polygon: lower herbicide
and fertiliser levels allow ruderal weed species to persist [80–82], and decreased levels of ploughing,
improve perennial weed survival [80], and germination rates of several species [83]. Field edges also
have increased numbers of more competitive species due to spill over of species from neighbouring
habitats [84]. Additionally, field boundary hedges can trap, or redirect seed dispersal [85,86] and
harbour animals, which are pollinators, consumers, or dispersers of weeds (reviewed in Petit, et al. [87].
Some spatial models account for the importance of field edges, whilst others choose to negate this
issue by focussing on the central part of the field.
5.4. Interactions
Currently, where models of within-field weed distributions consider external processes they
frequently do so in isolation. However, whilst the factors discussed in this section each play a role in
individually shaping the within-field distribution of weeds, they can also act in combination, both
synergistically and antagonistically, which could be considered in spatial models. Several environmental
variables may be correlated, making it difficult to disassociate one from another, for example, variation
in weed distribution due to local topographic variations can be linked to multiple soil factors [70].
Similarly, ecological factors may interact in spatially explicit ways, for example, predators affect weed
distribution and density, just as weed distribution and density also influences predator population
density [88]. In addition, environmental and ecological drivers of spatial heterogeneity may interact,
as not only are weeds affected by soil characteristics [89], but they can themselves, in turn, affect the
quality of the soil [90–92]. Persistent weedy patches will influence both future weed invasions, and crop
yields. Often, when a patch forms there will be a large seed return in following years, leading to patch
longevity [70]. In addition, patches can become self-regulating when the weeds produce secondary
metabolites (allelochemicals), which influence the germination, growth, survival, and reproduction of
other plant species [93].Agronomy 2020, 10, 1044 9 of 18
6. Sensitivity
Agronomy 2020,Analysis
10, x FOR PEER REVIEW 9 of 18
6.1.6.Uncertainty
Sensitivity Analysis
Sensitivity analysis determines how different variables contribute to the model’s overall uncertainty,
6.1. Uncertainty
where spatial models come with a unique set of considerations. For example, in CA models, both
the scaleSensitivity analysis determines how different variables contribute to the model’s overall
and the size of the lattice may affect the simulation results [33]. In addition, the optimum
uncertainty, where spatial models come with a unique set of considerations. For example, in CA
size for sub-populations in CA-P analyses can be influenced by many biotic factors, meaning that
models, both the scale and the size of the lattice may affect the simulation results [33]. In addition,
sensitivity analysis in relation to lattice planning is essential. Somerville et al. [32] also demonstrated
the optimum size for sub-populations in CA-P analyses can be influenced by many biotic factors,
thatmeaning
decisions onsensitivity
that cell size need to be
analysis species to
in relation specific.
lattice planning is essential. Somerville et al. [32] also
demonstrated that decisions on cell size need to be species specific.
6.2. Error
6.2. Errorin estimations of weed density and distribution in a weed model can be divided into two
Error
types: estimating that weeds
Error in estimations are worse
of weed density than
andthey truly are
distribution in (a type model
a weed one error),
can beand estimating
divided into twoweeds
are types:
not asestimating
bad as they thatare in reality
weeds (a type
are worse thantwo
theyerror). Type
truly are one one
(a type errors mean
error), andweed control
estimating measures
weeds
willare
be not as bad as they
implemented are in
when reality
they are (a
nottype two error).
needed, wastingTypetime
one and
errors mean weed
money. Type control measures
two errors mean that
will will
weeds be implemented
incorrectly when
not bethey are not needed,
controlled, wasting
possibly time and
resulting money. Type crop
in unexpected two errors mean
losses, andthat
a larger
weeds will incorrectly not be controlled, possibly resulting in unexpected
weed seed bank [94]. Error can be reduced by increasing the frequency, size, and analysis of crop losses, and a larger
the weed
weed seed bank [94]. Error can be reduced by increasing the frequency, size, and analysis of the weed
images, or reducing the time between image capture and spraying. Changes in threshold calculations
images, or reducing the time between image capture and spraying. Changes in threshold calculations
will affect the balance between type one and type two errors, for example in Figure 2, when only the
will affect the balance between type one and type two errors, for example in Figure 2, when only the
darker red area is sprayed this uses the least herbicide, whereas spraying the yellow area is a more
darker red area is sprayed this uses the least herbicide, whereas spraying the yellow area is a more
conservative
conservative approach
approachto to weed control.The
weed control. The yellow
yellow areaarea
reducesreduces type
type two two(missed
errors errors weeds),
(missedbutweeds),
butincreases
increasestype
typeone errors (wasted spray). Both types of errors can occur within the same field, and field,
one errors (wasted spray). Both types of errors can occur within the same
andmay
mayincur
incur different levels
different levels of concern
of concern to thetoagronomist.
the agronomist.
Figure
Figure2. Diagrammatic
2. Diagrammaticrepresentation
representation of ofusing
usinga awithin-field
within-field spatial
spatial weed
weed modelmodel for patch
for patch spraying.
spraying.
Area (A)(A)
Area is aiszone of actual
a zone of actualor or
predicted
predictedweeds.
weeds.AAbuffer
bufferzone
zone (B)
(B) allows for the
allows for thespread
spreadofofthe
thepatch
patch into
the into the current
current season.season. The field
The field is thenis then divided
divided intointo grids
grids ofof a manageable size—often
a manageable size—often the thewidth
width ofof the
theboom—and
spray spray boom—and all cells
all grid grid cells
thatthat contain
contain thethe patch
patch andbuffer
and bufferzones
zones will
will be
be sprayed
sprayed(cells within
(cells within the
darkthered
dark redUnfortunately,
line). line). Unfortunately, it is possible
it is possible that weeds
that weeds havehave
beenbeen dispersed
dispersed (areas
(areas C and
C and D),D),
or or
entered
entered the field from the margin (E). Spatio-temporal weed models can
the field from the margin (E). Spatio-temporal weed models can predict the future distribution of predict the future
distribution of weeds or identify the probability of weed densities exceeding certain thresholds in
weeds or identify the probability of weed densities exceeding certain thresholds in particular locations
particular locations (shaded grey areas). Then we may be able to identify “weed vulnerable zones”
(shaded grey areas). Then we may be able to identify “weed vulnerable zones” and spray accordingly
and spray accordingly (cells within the light yellow line). However, the seeds entering the field from
(cells within the light yellow line). However, the seeds entering the field from the margin (area E)
the margin (area E) would only be sprayed if margins were included in the predictive modelling, and
would only be sprayed if margins were included in the predictive modelling, and a very conservative
a very conservative approach was used.
approach was used.Agronomy 2020, 10, 1044 10 of 18
6.3. Stochasticity
Many of the factors affecting weed distribution are partly random in nature, making their full
spatial assessment at least as difficult and time consuming as field-wide weed counts. To help account
for this variability, stochastic processes are often used in simulations. However, the agroecosystem is a
complex interaction of biological mechanisms, agricultural activities, and environmental variables.
Stochastic simulations are different from deterministic simulations, because they use probabilities
to govern their biological processes. Stochastic parameterisations can be used in spatial, or non-spatial
simulation models, but their incorporation into CA simulations means that a range of results will be
found at the same time at different locations within the lattice, even when the starting population is
uniform. Spatial variability in the starting population has been used in deterministic models to achieve
reliable results in short run simulations [95]. However, at densities of less than one weed per cell,
the rounding in deterministic models will extinguish sub-populations earlier than in stochastic models.
7. Use-Cases and Future Directions
Models of within-field weed spatial distributions have been developed for several purposes,
currently, the targeted application of herbicides is a predominant use. However, more varied uses also
exist, with both farmers and scientists as the targeted end-user.
7.1. Targeted Weed Treatments
Integrated weed management (IWM) is promoted, both as a way to reduce the reliance on
herbicides [96], and as a method to slow herbicide resistance evolution [97]. Spatial modelling can be
useful in promoting IWM; targeted spraying of both current, and pre-emergent weeds is a growing
area of research into lowering herbicide use. Improvements in static spatial modelling should help
build better weed maps for spraying current and pre-emergent weeds, with additional advantages
provided from using spatio-temporal models to predict weed numbers and crop yields in future years.
Both real time assessments and interpolated weed maps are useful for the design of within-field
spray maps for herbicide applications, particularly when they incorporate buffer zones to account for
potential patch spread. Spatio-temporal models can be used to predict “weed-vulnerable zones” and
patch persistence [39]. The resulting weed maps may be useful to generate predictive spray maps for
use with pre-emergence herbicides, where it is not possible to spray based on observations of weed
distributions alone (Figure 2).
Each herbicide spray map needs to consider not only the flora, but also the capacity of the sprayer.
The technology development in designing weed maps must be linked closely to the capacity of farmers
to access appropriate sprayers. Sprayers with more nozzles across the boom, and graduated control
of each nozzle, can implement more complex spray maps [13], although large herbicide savings can
still be achieved with limited sprayer refinement [4]. In addition, sprayers may be developed with
the capacity to apply multiple herbicides in a single operation, thereby further refining the weed
modelling criteria.
7.2. Herbicide Resistance and Integrated Weed Management+
Outcrossing annual weed species producing high numbers of seeds are more susceptible to
herbicide resistance, with high levels of herbicide resistance confined to a small number of species [98].
However, changing climatic conditions [99], new farming practices [100,101], legislative changes [102],
and higher levels of international trade [103] will expose a new cohort of different weed species to
different herbicide regimes. Herbicide resistance selection pressure is dynamic, and, in many instances,
worsening [102,104].
Simulating spatial movement of HR genes was reviewed in Bagavathiannan and Norsworthy [105],
where they comment on the importance of within field spatial differences in both biology and
environment. Knowledge of the location and movement of HR genes within fields is important,Agronomy 2020, 10, 1044 11 of 18
as diversity in the within-field management of weeds (integrated weed management) is the best
method to reduce herbicide resistance [97]. Typically, spatio-temporal models predict slower rates
of HR evolution (see Section 3.2), even for a single gene, when compared to (non-spatial) temporal
models [33,34,106]. In contrast, high concentrations of HR weeds develop in localised patches in CA
simulations [33].
Several HR genes are either partially, or fully recessive [107], meaning that homozygous HR weeds
have a greater chance of surviving herbicide applications. In addition, two genetically different, initially
rare HR genes, causing multiple resistance (in the same weed plant), are unlikely to spontaneously
occur near each other within a field [28,108]. CA models will, by definition, develop patches of weeds
with the same form of HR (giving rise to more homozygous HR weeds) and simultaneously maintain
some separation between different forms of HR genes, giving rise to less multiple HR weeds [109].
Therefore, CA models examining either semi- or fully-recessive HR genes [109,110], or the development
of a multiple HR population [28], will give markedly different results to non-spatial simulations, which
consider all weeds to be part of a freely interbreeding non-spatially defined, weed population.
Much of the earlier work on the spatial dispersal of herbicide resistance included a fitness cost
within the HR weeds [38,109], and examined the effects of rotating herbicides or incorporating areas or
times of reduced spraying. However, the coupling of HR and a fitness cost is evolutionarily selected
against, and the permanence of any fitness cost is now questioned [111]. The modelling of HR genes is
useful in identifying the causes of HR, and best integrated weed management strategies. Modelling can
help in the fight against present, and future outbreaks of herbicide resistance.
7.3. Cost-Benefit Analyses of Weed Management
Spatial models of weed populations can also be used to investigate the economic viability of
different management practices. For example, Audsley [112] used a spatially explicit model of
Avena fatua and A. myosuroides to determine the weed density at which patch spraying becomes more
cost effective than broadcast spraying. Similarly, González-Díaz et al. [30] used a spatially explicit
bio-economic model to determine that integrated weed management strategies are more cost effective
than strategies with individual management tactics, such as herbicide application or crop rotation
change, in controlling Lolium rigidum populations. Bio-economic models can be of value in advisory
services to visualise long-term consequences, which are not immediately evident for farmers.
7.4. Biocontrol, Invasive Species and Genetic Spread
Designing an invasive spread model to identify or track the spread of a new species, or the spread
of new genetic material, is important in both farming and biosecurity. Spatio-temporal models of
expansion from an initial site are simpler to model than whole fields of weeds, as the spread will
typically be radially outwards from the initial source. This limited direction of spread means that both
CA-E and non-CA models are suitable [35,41,113], although CA-P models have also been used [28].
Predictions of weed densities and distribution into the future are useful for maintaining biological
weed control agents. The persistence of patches, as well as their density, size, and distribution,
can all affect the persistence of biological weed control agents [21,88,114]. In addition, interactions
occur, with reductions in seed predator numbers, tillage [115], and increased predation rates by weed
patchiness [21,116,117]. Herbivory can affect both weed seed production, and seed dispersal, adding
further complexity to models of weeds under biocontrol [118].
Simulated investigations into alternate strategies of biological control can investigate different
biological control agents, release dates, and agent clumping strategies. However, the ideal choice can
be dependent on whether spatial containment or population minimization is the aim of the control
agent [119], and whether type one or type two errors are of greater concern [94].Agronomy 2020, 10, 1044 12 of 18
7.5. Future Directions
In addition to the wide scope for using weed spatial and spatio-temporal models to answer
emerging questions in weed ecology and management, we also see potential for the continuing
improvement of the models themselves. Ongoing work in image analysis and machine learning is
providing new means of real time weed detection, whilst novel geostatistical methods are enabling
more reliable ways of producing accurate weed maps. In turn, these weed maps can provide improved
inputs for CA models with the potential for multiple realisations of the same map, according to the
uncertainty associated with it. In the area of model mechanics, there are increasing numbers of more
detailed data sets available for parameterisation. Improvements in the understanding of weed species
ecology and responses to management can also be incorporated into future models to allow more
accurate predictions to be made. In addition, ongoing dynamics in the farming sector (Section 7.2) will
continue to provide impetus for future model developments.
8. Conclusions
Static spatial within-field weed models have many uses for growers, including tracking weeds,
and targeting herbicide applications. The greatest range of development in weed modelling is currently
observed for the spatio-temporal models. New models have focused on a wide range of problematic
weed species and farming systems and, as such, parameterisation of the pertinent ecological processes
has been widespread. Many modellers choose to use a CA approach (particularly CA-P) as this not
only allows for simulation of spatially explicit processes, but maintains tractability of the model and
provides output at a scale suitable for management decisions to be applied.
Traditionally, biological and ecological processes have formed the basis of spatio-temporal weed
models with particular emphasis on processes such as dispersal and competition, which become
particularly relevant when considering spatial distributions at the within-field scale. However, more
recently, increased consideration has been given to external drivers of weed distribution, including
the relationship between weeds and environmental properties, as well as anthropogenic drivers of
weed distributions.
We identified a number of key areas, in Section 7, for which the development of spatial weed
models will be key to advancing the understanding of weed biology and ecology, and also in terms of
allowing scientists to make recommendations about the best management practices to be implemented.
Author Contributions: Conceptualization, G.J.S.; methodology, investigation, resources, original draft preparation,
review and editing, supervision and project administration by all authors; funding acquisition, S.K.M. and M.S.
All authors have read and agreed to the published version of the manuscript.
Funding: H.M. and M.S. received funding from the European Union’s Horizon 2020 Research and Innovation
Programme under grant agreement No 727321. H.M. is partly supported by the Natural Environment Research
Council (NERC) and the Biotechnology and Biological Sciences Research Council (BBSRC) research programme
NE/N018125/1 LTS-M (ASSIST). G.J.S. and S.K.M. received funding from the Danish Innovation Fund through the
project RoboWeedMaPS (project 6150-00027B).
Acknowledgments: Thank you to subsequent employers of G.J.S.: Some work on this manuscript was undertaken
whilst she was employed, and under covid19 lockdown, at Rothamsted Research (UK), and later, at Lincoln
University (NZ).
Conflicts of Interest: The authors declare no conflict of interest, the funders had no role in the design of the study;
in the collection, analyses, or interpretation of resources; in the writing of the manuscript, or in the decision
to publish.Agronomy 2020, 10, 1044 13 of 18
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